The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

The 8. Theoreme. The 8. Proposition. Pyramids being like & hauing triangles to their bases, are in treble propor¦tion the one to the other, of that in which their sides of like proportion are.

SVppose that these pyramids whose bases are the triangles GBC and HEF and toppes, the poyntes A and D be like, and in like sort described, and let AB and DE be sides of like proportion. Then I say that the pyramis ABCG is to the pyramis DEFH in treble proportiō, of that in which the side AB is to the side DE. Make perfect the parallelipipedons, namely, the solides BCKL & EFXO. And foras∣much as the pyramis ABCG is like to the pyramis DEFH,* 1.1 therfore the angle ABC is equall to the angle DEF,* 1.2 & the

[illustration]

angle GBC to the angle HEF, and moreouer the angle ABG to the angle DEH, and as the line AB is to the line DE, so is the line BC to the line EF, and the line BG to the line EH. And for that as the line AB is to the line DE, so is the line BC to the line EF, and the sides about the equall an∣gles, are proportionall, therefore the parallelo∣gramme BM is like to the parallelogrāme EP: and by the same reason the pa∣rallelogramme BN is like

to the parallelogramme ER, and the parellelogramme BK is like vnto the parallelogramme EX. Wherefore the three parallelogrammes BM, KB and BN are like to the three paral∣lelogrammes EP, EX, and ER. But the three parallelogrammes BM, KB, and BN are equall and like to the three opposite parallelogrammes, and the three parallelogrammes EP, EX, and ER are equall and like to the three opposite parallelogrammes. Wherefore the pa∣rallelipipedons BCKL and EFXO are comprehended vnder playne superficieces like and equall in multitude. Wherefore the solide BCKL is like to the solide EFXO. But like paral¦lelipipedons are (by the 33. of the eleuenth) in treble proportion the one to the other of that in which side of like proportion is to side of like proportion. Wherefore the solide BCKL is to the solide EFXO in treble proportion of that in which the side of like proportion AB is to the side of like proportion DE. But as the solide BCKL is to the solide EFXO, so is the py∣ramis ABCG to the pyramis DEFH (by the 15. of the fifth) for that the pyramis is the sixth part of this solide: for the prisme, being the halfe of the parallelipipedon is treble to the pyramis. Wherefore the pyramis ABCG is to the pyramis DEFH in treble proportion of that in which the side AB is to the side DE. Which was required to be proued.

Corollary.

Hereby it is manifest that like pyramids hauing to their bases poligonon figures, are in treble proportion the one to the other, of that in which side of like proportion, is to side of like proportion.

For if they be deuided into pyramids hauing triangles to their bases (for like poligonō figures are deuided into like triangles, and equal in multitude, and the sides are of like propor∣tion) as one of the pyramids of the one, hauing a triangle to his base, is to one of the pyramids of the other, hauing also a triangle to his base, so also are all the pyramids of the one pyramis hauing triangles to their bases to all the pyramids of the other pyramis hauing also triangles to their bases. That is, the pyramis hauing to his base a poligonō ••igure, to the pyramis hauing also to his base a poligonō ••igure. But a pyramis hauing a triangle to his base, is to a pyramis ha¦uing also a triangle to his base, & being like vnto it, in treble proportiō of that in which side of like proportiō is to side of like proportiō. Wherfore a pyramis hauing to his base a poligonō figure, is to a pyramis hauing also a poligonon figure to his base, the sayd pyramids being like the one to the other, in treble proportion of that in which side of like proportion is to side of like proportion. Likewise Prismes and sided columnes, being set vpon the bases of those pyramids,* 1.3 and vnder the same altitude (forasmuch as they are equemultiplices vnto the pyramids, namely, triples, by the corollary of the 7. of this booke) shal haue the ••ormer porportion that the pyramids haue, by the 15, of the fifth, and therefore they shall be in treble proportion of that in which the sides of like propor∣tion are.

Notes

• * 1.1

  Construction.
• * 1.2

  Demonstra∣tion.
• * 1.3

  An addition by Campane and Flussas.